CH11-1: Euler’s Formula Geometry B-CH11 2 Surface Area …€¦ · · 2018-04-26CH11-1: Euler’s Formula Background: This Swiss guy, ... Segment that is formed by the intersection

8/23/2011

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MRS . KUMMER

FALL , 201 1 - 2012

Geometry B-CH11 Surface Area and Volume

CH11-1: Euler’s Formula

Background: This Swiss guy, Leonhard Euler lived from 1707 to 1783 doing

most of his work in Berlin, Germany.

In 1735 through 1766 he went totally blind. He dictated his

formulas and mathematical papers to

an assistant and did most of his calculations in his head! His formulas

and work served as a basis of

advanced math topics like differential equations (math beyond Calculus!)

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Vocabulary:

Euler developed a formula to help analyze various

polyhedrons.

Polyhedron: a 3D shape with a surface of a

polygon (ex. Volleyball).

Face: The polygon is called the face.

Edge: Segment that is formed by the intersection of two faces.

Vertex: a point where 3 or more edges intersect.

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Euler’s Formula:

F + V = E +2

F = # Faces

V=# Vertices

E = # Edges

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Ex.1 Use Euler’s Formula to find the missing number for the polyhedron.

#Faces:

6

#Edges:

12

#Vertices=?

F+V = E +2

6+V = 12+2

6+V = 14

-6 = -6

V = 8

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Now, you do

#s 1-9 on page 601!

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CH11-2 Surface Areas of Prisms and Cylinders

Surface Area (SA): Sum of the areas of all the faces of a 3D object.

Lateral Area (LA): Sum of the areas of all the faces EXCEPT THE TOP AND THE BOTTOM.

The word lateral means “Side.”

Since it is still a calculation of area, the units for Surface Area and Lateral Area are still squared units.

Ex. m2, cm2, in2

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11-2 Surface Areas of Prisms and Cylinders

Let’s go make a model!

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CH11-2: Surface Area of Prisms and Cylinders

Lateral Area of a Prism

LA = Sum of areas of all sides

h

9


Surface Area of a Prism

SA = LA +2∙Ab

LA = Sum of areas of sides

Ab = Area of Base Shape

h

10


Ex.1 Find a) the lateral area and b) the surface area of each prism. DON’T FORGET YOUR UNITS!

LA = Sum of area of sides

What shape are the sides?

Rectangles

What can we use to

find missing sides??

c2 = a2 + b2

c2 = 52 + 82

c2 = 25 + 64

c2 = 89

c = 9.43 in.

18 in.

10 in.h = 8

in.

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Now, we can find LA:

LA =Sum of areas of all sides

What side shape do we have?

Rectangle

Area of Rectangle = b∙h

Area of rectangle 1 = 10∙18

Area of rectangle 2= 9.4∙18

Area of rectangle 3 = 9.4∙18

LA=Area1+Area2+Area3

LA = 180+169.2+169.2

LA = 518.4 in2

18 in.

10 in.h = 8

in.

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Now, we can find SA:

SA = LA +2∙Ab



What base shape do we have?

Triangle

Area of Triangle = ½ ∙b∙h

Area of Triangle = ½ ∙(10in)(8in)

Area of Triangle = 40 in2

SA = LA + 2∙Ab

SA = 518.4 + 2(40) in2

SA = 598.4 in2

18 in.

10 in.h = 8

in.

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Now, you do EVENS

8, 10, & 12

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Lateral Area of a Cylinder

LA = 2∙π ∙ r ∙ h

LA = Lateral Area

r = radius of circle

h= height

Surface Area of a Cylinder

SA = LA +2 ∙ π ∙ r2

h

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Surface Area of a Prism

SA = LA +2∙Ab



h

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Ex.2 Find the surface area of each cylinder in terms of π.

SA = 2∙π ∙ r ∙ h ++++ 2 ∙ π ∙ r2

SA = 2∙π ∙ 1m ∙ 3m+2 ∙ π ∙ (1m)2

SA = 6π +2π

SA = 8π

1m

3m

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Now, you do EVENS

2,4,6,14

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CH11-3: Surface Area of Pyramids and Cones

Lateral Area of a Pyramid

LA = ½∙llll····PBase

LA = Lateral Area

LLLL = slant height

PBase=Perimeter of Base Shape

h

19

llll


Surface Area of a Pyramid

SA = LA + ABase

LA = Lateral Area

Abase = Area of Base Shape

h

20

llll

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Ex.1 Find a) the lateral area

and b) the surface area of the

shape provided.

First, the Lateral Area

What shape is the base?

Square, so

LA = ½∙llll····Pbase

LA = ½∙llll····(12+12+12+12)

How do we find slant height, llll?

21 ft

21

llll

12 ft12 ft


Yes! Pythagorean Theorem

c2 = a2+b2

c2 = 212+62

c2 = 441+36

c2 = 477

√c2 = √477

c= 21.84

Now, we can find Lateral Area, LA

21 ft

22

llll

12 ft12 ft


LA = ½∙llll····(12+12+12+12)

And we now know c=llll=21.84

LA = ½∙21.84····(12+12+12+12)

LA = 524.2 ft2


Square

SA = LA + Abase

SA = 524.2 + (12)2

SA = 668.2 ft2

21 ft

23

llll

12 ft12 ft


Now, you do 7, 9, and 11

21 ft

24

llll

12 ft12 ft

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Lateral Area of a Cone

LA = π∙r∙llll

LA = Lateral Area

LLLL = slant height

r=radius of base

25

r

llll


Surface Area of a Cone

SA = π∙r∙llll+πr2

SA = Surface Area

LLLL = slant height

r=radius of base

26

r

llll


Now, find Lateral Area

c2=a2+b2

c2=202+92

c2=400+81

c2=481

√c2=√481

c=21.9

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9cm

20cmllll




LA = π∙r∙llll

LA = π∙(9)∙(21.9)

LA = 618.9 cm2

SA = π∙r∙llll+πr2

SA = π∙9∙(21.9)+π92

SA = 197.1π+81π

SA = 278.1π cm2

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9cm

20cmllll

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Now, you do 1, 3, and 5

21 ft

29

llll

12 ft12 ft

CH11-4 Volumes of Prisms & Cylinders

Background: Often, it is useful to know the amount of the inside of a 3D shape. For example, out in

Milan Dragway, there are large plastic cylinders for

recycling used oil.

Vocabulary:

Volume, V: Product of any 3 dimensions.

Measures an objects INTERIOR PLUS DEPTH and

has cubed units. Ex. m3, cm3, ft3

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Volume of a Cylinder

V=π•r2•h

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r

h


Ex.1 Find the volume of each cylinder to the nearest tenth.

V=π•r2•h

V = π•(2m)2•(2m)

V = 25.1 m3

4m

2m

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Now, you do

EVENS 2,4, 6

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CH11-4 Volumes of Prisms & Cylinders34

Volume of a Prism

V=Ab•h

Ab=Area of Base Shape

h=height of prism

18

in.

10

in.

h =

8 in.

CH11-4: Volume of Prisms & Cylinders

Ex.1 Find the volume of each prism.

First, what shape is the base?

Triangle, so

Ab = ½∙b∙h

Ab = ½∙(10in)∙(8in)

Ab = 40 in2

V= Ab•h

V = (40in2) •(18in)

V = 720 in3

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18 in.

10 in.h = 8

in.


Now, you do

EVENS 8 -14

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CH11-5: Volumes of Pyramids and Cones

Volume of a Pyramid

V = 1/3∙Ab∙h


h= height of pyramid

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h


Ex.1 Find the volume of each

pyramid.


Square, so

Ab = s2

Ab = (12cm)2

Ab = 144cm2

V = 1/3∙Ab∙h

V = 1/3∙(144cm2)∙(21cm)

V = 1008 cm3

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21 ft

llll

12 ft12 ft


Now, you do #S,

1,3,5,13&15

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Volume of a Cone

V = 1/3∙π∙r2∙h

r = radius

h=height of cone

(use Pythagorean Theorem or Trig to find h)

40

r

llll

h

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Ex.2 Find the Volume of the cone.

V = 1/3∙π∙r2∙h

V = 1/3∙π∙(9cm)2∙(20cm)

V = 1696.5 cm3

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9cm

20cm

llll


Now, you do #s 7,9, & 11

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11-6 Surface Areas and Volumes of Spheres43

Surface Area of a Sphere

SA= 4∙π∙r2

r = radius r

Volume of a Sphere

V= 4/3∙π∙r3

r = radius

11-6 Surface Areas and Volumes of Spheres

Ex.1 Find a) the Surface Area and b) Volume of the sphere. Round your answers to the nearest tenth.

a)

SA= 4∙π∙r2

SA= 4∙π∙(5m)2

SA= 314.2 m2

b)

V= 4/3∙π∙r3

V= 4/3∙π∙(5m)3 V= 1.3333∙π∙(5m)3

V = 523.6 m3

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10 m

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CH11-6: Surface Areas and Volumes of Spheres

Now, you do EVENS (2-18)!

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11-7 Areas & Volumes of Similar Solids

Background: Sometimes, you don’t have all the dimensions of all sides for your shapes. So, if you

know the surface areas or volumes, you can make a

proportion to figure it out.

Vocabulary:

Surface Area: The total of any shape rolled out flat.

Proportion: Two ratios set equal to each other.

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SA1 = a2

SA2 b2

V1 = a3

V2 b3

a

b = similarity ratio

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a

b

How To Use It:

Ex.1 For each pair of similar figures, find the similarity ratios of the smaller to the larger shape.

SA1 = a2

SA2 b2

9 = a2

16 b2

√ 9 = √a2

√16 √b2

3=a

4 b

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SA = 9SA = 16

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Now, you do ODDS

1-11

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Woooohoooo! We’re done with CH11!!!!!!

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Documents

CH11-1: Euler’s Formula Geometry B-CH11 2 Surface Area …€¦ · · 2018-04-26CH11-1: Euler’s Formula Background: This Swiss guy, ... Segment that is formed by the intersection